(a) Prove that the eigenvalues of a real symmetric positive-definite matrix $A$ are all positive.

Let $\lambda$ be a (real) eigenvalue of $A$ and let $\mathbf{x}$ be a corresponding real eigenvector. That is, we have
\[A\mathbf{x}=\lambda \mathbf{x}.\]
Then we multiply by $\mathbf{x}^{\trans}$ on left and obtain
\begin{align*}
\mathbf{x}^{\trans}A\mathbf{x}&=\lambda \mathbf{x}^{\trans} \mathbf{x}\\
&=\lambda ||\mathbf{x}||^2.
\end{align*}

The left hand side is positive as $A$ is positive definite and $\mathbf{x}$ is a nonzero vector as it is an eigenvector.
Since the length $||\mathbf{x}||^2$ is positive, we must have $\lambda$ is positive.
It follows that every eigenvalue $\lambda$ of $A$ is real.

(b) Prove that if eigenvalues of a real symmetric matrix $A$ are all positive, then $A$ is positive-definite.

Note that a real symmetric matrix is diagonalizable by an orthogonal matrix.

Let $\mathbf{x}$ be an arbitrary nonzero vector in $\R^n$.
Since $A=QDQ^{\trans}$ (remark that $Q^{-1}=Q^{\trans}$), we have
\[\mathbf{x}^{\trans} A\mathbf{x}=\mathbf{x}^{\trans}QDQ^{\trans}\mathbf{x}.\]
Putting $\mathbf{y}=Q^{\trans}\mathbf{x}$, we can rewrite the above equation as
\[\mathbf{x}^{\trans} A\mathbf{x}=\mathbf{y}^{\trans}D\mathbf{y}.\]

By assumption eigenvalues $\lambda_i$ are positive.
Also, since $\mathbf{x}$ is a nonzero vector and $Q$ is invertible, $\mathbf{y}=Q^{\trans}\mathbf{x}$ is not a zero vector.
Thus the sum expression above is positive, hence $\mathbf{x}^{\trans} A\mathbf{x}$ is positive for any nonzero vector $\mathbf{x}$.
Therefore, the matrix $A$ is positive-definite.

Transpose of a Matrix and Eigenvalues and Related Questions
Let $A$ be an $n \times n$ real matrix. Prove the followings.
(a) The matrix $AA^{\trans}$ is a symmetric matrix.
(b) The set of eigenvalues of $A$ and the set of eigenvalues of $A^{\trans}$ are equal.
(c) The matrix $AA^{\trans}$ is non-negative definite.
(An $n\times n$ […]

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Show that eigenvalues of a Hermitian matrix $A$ are real numbers.
(The Ohio State University Linear Algebra Exam Problem)
We give two proofs. These two proofs are essentially the same.
The second proof is a bit simpler and concise compared to the first one.
[…]

Questions About the Trace of a Matrix
Let $A=(a_{i j})$ and $B=(b_{i j})$ be $n\times n$ real matrices for some $n \in \N$. Then answer the following questions about the trace of a matrix.
(a) Express $\tr(AB^{\trans})$ in terms of the entries of the matrices $A$ and $B$. Here $B^{\trans}$ is the transpose matrix of […]

There is at Least One Real Eigenvalue of an Odd Real Matrix
Let $n$ be an odd integer and let $A$ be an $n\times n$ real matrix.
Prove that the matrix $A$ has at least one real eigenvalue.
We give two proofs.
Proof 1.
Let $p(t)=\det(A-tI)$ be the characteristic polynomial of the matrix $A$.
It is a degree $n$ […]